Additive Representations of Natural Numbers

Abstract

We establish an improved lower bound for a weighted counting function of representations of n as the sum of a prime and a square-free number and provide several applications for this bound. These applications are generalisations of a result of Dudek [4], which states that every natural number greater than 2 may be written as the sum of a prime and a square-free number.

On the sum of a prime and a square-free number with divisibility conditions

“…Then every n ≥ 36 can be expressed as the sum of a prime and a square-free number η > 1 with (η, k) = 1. These results improve on recent work of Francis and Lee [FL20], who only considered the case where k is prime.…”

“…Remark. The computations for k prime were also performed by Francis and Lee [FL20]. However, we thought it would be instructive to reproduce their result.…”

“…In particular, R k (n) is the number of representations of n as the sum of a prime and a square-free number coprime to k. Francis and Lee [FL20] recently analysed the case where k is prime but we consider more general k here.…”

“…As motivated in the previous section, we now provide bounds for R d,e (n). This will be done using a method similar to that in [Dud17] and [FL20]. Here,…”

“…We employ an algorithm similar to [FL20]. We first compute the set S of squarefree numbers less than or equal to 1.6 • 10 8 , excluding 1.…”